Suppose you've got an argument with one premise and one conclusion, and you think its valid. Call the premise p and the conclusion q. Plausibly, constraints on rational belief follow: in particular, you can't rationally have a lesser degree of belief in q than you have in p.
The natural generalization of this to multi-premise cases is that if p1...pn|-q, then your degree of disbelief in q can't rationally exceed the sum of your degrees of disbelief in the premises.
FWIW, there's a natural generalization to the multi-conclusion case too (a multi-conclusion argument is valid, roughly, if the truth of all the premises secures the truth of at least one conclusion). If p1...pn|-q1...qm, then the sum of your degrees of disbelief in the conclusions can't rationally exceed the sum of your degrees of disbelief in the premises.
What I'm interested in at the moment is to what extent this sort of connection can be extended to non-classical settings. In particular (and connected with the last post) I'm interested in what the supervaluationist should think about all this.
There's a fundamental choice to be made at the get-go. Do we think that "degrees of belief" in sentences of a vague language can be represented by a standard classical probability function? Or do we need to be a bit more devious?
Let's take a simple case. Construct the artificial predicate B(x), so that numbers less than 5 satisfy B, and numbers greater than5 fail to satisfy it. We'll suppose that it is indeterminate whether 5 itself is B, and that supervaluationism gives the right way to model this.
First observation. It's generally accepted that for the standard supervaluationist
Given this and the constraints on rational credence mentioned earlier, we'd have to conclude that my credence in B(5)&~Det(B(5)) must be 0. I have credence 0 in absurdity; and the degree of disbelief in the conclusion of this valid argument (1) must not exceed the degree of disbelief in its premises.
Let's think that through. Notice that in this case, my credence in ~Det(B(5)) can be taken to be 1. So given minimal assumptions about the logic of credences, my credence in B(5) must be 0.
A parallel argument running from ~B(5)&~Det(~B(5))|-absurdity gives us that my credence in ~B(5) must be 0.
Moreover, supervaluational entails all classical tautologies. So in particular we have the validity: |-B(5)v~B(5). The standard constraint in this case tells us that rational credence in this disjunction must be 1. And so, we have a disjunction in which we have credence 1, each disjunct of which we have credence 0 in. (Compare the standard observation that supervaluational disjunctions can be non-prime: the disjunction can be true when neither disjunct is).
This is a fairly direct argument that something non-classical has to be going on with the probability calculus. One move at this point is to consider Shafer functions (which I know little about: but see here). Now maybe that works out nicely, maybe it doesn't. But I find it kinda interesting that the little constraint on validity and credences gets us so quickly into a position where something like this is needed if the constraint is to work. It also gives us a recipe for arguing against standard supervaluationism: argue against the Shafer-function like behaviour in our degrees of belief, and you'll ipso facto have an argument against supervaluationism. For this, the probablistic constraint on validity is needed (as far as I can see): for its this that makes the distinctive features mandatory.
I'd like to connect this to two other issues I've been working on. One is the paper on the logic of supervaluationism cited below. The key thing here is that it raises the prospect of p&~Dp|-absurdity not holding, even for your standard "truth=supertruth" supervaluationist. If that works, the key premise of the argument that forces you to have degree of belief 0 in both an indeterminate sentence 'p' and its negation goes missing.
Maybe we can replace it by some other argument. If you read "D" as "it is true that..." as the standard supervaluationist encourages you to, then "p&~Dp" should be read "p&it is not true that p". And perhaps that sounds to you just like an analytic falsity (it sure sounded to me that way); and analytic falsities are the sorts of things one should paradigmatically have degree of belief 0 in.
But here's another observation that might give you pause (I owe this point to discussions with Peter Simons and John Hawthorne). Suppose p is indeterminate. Then we have ~Dp&~D~p. And given supervaluationism's conservativism, we also have pv~p. So by a bit of jiggery-pokery, we'll get (p&~Dp v ~p&~D~p). But in moods where I'm hyped up thinking that "p&~Dp" is analytically false and terrible, I'm equally worried by this disjunction. But that suggests that the source of my intuitive repulsion here isn't the sort of thing that the standard supervaluationist should be buying. Of course, the friend of Shafer functions could just say that this is another case where our credence in the disjunction is 1 while our credences in each disjunct is 0. That seems dialectically stable to me: after all, they'll have *independent* reason for thinking that p&~Dp should have credence 0. All I want to insist is that the "it sounds really terrible" reason for assigning p&~Dp credence 0 looks like it overgeneralizes, and so should be distrusted.
I also think that if we set aside truth-talk, there's some plausibility in the claim that "p&~Dp" should get non-zero credence. Suppose you're initially in a mindset where you should be about half-confident of a borderline case. Well, one thing that you absolutely want to say about borderline cases is that they're neither true nor false. So why shouldn't you be at least half-confident in the combination of these?
And yet, and yet... there's the fundamental implausibility of "p&it's not true that p" (the standard supervaluationist's reading of "p&~Dp") having anything other than credence 0. But ex hypothesi, we've lost the standard positive argument for that claim. So we're left, I think, with the bare intuition. But it's a powerful one, and something needs to be said about it.
Two defensive maneuvers for the standard supervaluationist:
(1) Say that what you're committed to is just "p& it's not supertrue that p". Deny that the ordinary concept of truth can be identified with supertruth (something that as many have emphasized, is anyway quite plausible given the non-disquotational nature of supertruth). But crucially, don't seek to replace this with some other gloss on supertruth: just say that supertruth, superfalsity and gap between them are appropriate successor concepts, and that ordinary truth-talk is appropriate only when we're ignoring the possibility of the third case. If we disclaim conceptual analysis in this way, then it won't be appropriate to appeal to intuitions about the English word "true" to kick away independently motivated theoretical claims about supertruth. In particular, we can't appeal to intuitions to argue that "p&~supertrue that p" should be assigned credence 0. (There's a question of whether this should be seen as an error-theory about English "truth"-ascriptions. I don't see it needs to be. It might be that the English word "true" latches on to supertruth because supertruth what best fits the truth-role. On this model, "true" stands to supertruth as "de-phlogistonated air" according to some, stands to oxygen. And so this is still a "truth=supertruth" standard supervaluationism.)
(2) The second maneuver is to appeal to supervaluational degrees of truth. Let the degree of supertruth of S be, roughly, the measure of the precisifications on which S is true. S is supertrue simpliciter when it is true on all the precisifications, i.e. measure 1 of the precisifications. If we then identify degrees of supertruth with degrees of truth, the contention that truth is supertruth becomes something that many find independently attractive: that in the context of a degree theory, truth simpliciter should be identified with truth to degree 1. (I think that this tendancy has something deeply in common with the temptation (following Unger) to think that nothing that nothing can be flatter than a flat thing: nothing can be truer than a true thing. I've heard people claim that Unger was right to think that a certain class of adjectives in English work this way).
I think when we understand the supertruth=truth claim in that way, the idea that "p&~true that p" should be something in which we should always have degree of belief 0 loses much of its appeal. After all, compatibly with "p" not being absolutely perfectly true (=true), it might be something that's almost absolutely perfectly true. And it doesn't sound bad or uncomfortable to me to think that one should conform one's credences to the known degree of truth: indeed, that seems to be a natural generalization of the sort of thing that originally motivated our worries.
In summary. If you're a supervaluationist who takes the orthodox line on supervaluational logic, then it looks like there's a strong case for a non-classical take on what degrees of belief look like. That's a potentially vulnerable point for the theory. If you're a (standard, global, truth=supertruth) supervaluationist who's open to the sort of position I sketch in the paper below, prima facie we can run with a classical take on degrees of belief.
Let me finish off by mentioning a connection between all this and some material on probability and conditionals I've been working on recently. I think a pretty strong case can be constructed for thinking that for some conditional sentences S, we should be all-but-certain that S&~DS. But that's exactly of the form that we've been talking about throughout: and here we've got *independent* motivation to think that this should be high-probability, not probability zero.
Now, one reaction is to take this as evidence that "D" shouldn't be understood along standard supervaluationist lines. That was my first reaction too (in fact, I couldn't see how anyone but the epistemicist could deal with such cases). But now I'm thinking that this may be too hasty. What seems right is that (a) the standard supervaluationist with the Shafer-esque treatment of credences can't deal with this case. But (b) the standard supervaluationist articulated in one of the ways just sketched shouldn't think there's an incompatibility here.
My own preference is to go for the degrees-of-truth explication of all this. Perhaps, once we've bought into that, the "truth=degree 1 supertruth" element starts to look less important, and we'll find other useful things to do with supervaluational degrees of truth (a la Kamp, Lewis, Edgington). But I think the "phlogiston" model of supertruth is just about stable too.
[P.S. Thanks to Daniel Elstein, for a paper today at the CMM seminar which started me thinking again about all this.]